Axiom of extensionality

The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as the Zermelo–Fraenkel set theory. The axiom defines what a set is.

The logical term was introduced to set theory in 1893, Gottlob Frege attempted to use this idea of an extension formally in his Basic Laws of Arithmetic (German: Grundgesetze der Arithmetik), where, if <math>F</math> is a predicate, its extension (German: Umfang) <math>\varepsilon F</math>, is the set of all objects satisfying <math>F</math>. For example if <math>F(x)</math> is "x is even" then <math>\varepsilon F</math> is the set <math>\{ \cdots , -4, -2, 0, 2, 4, \cdots \} </math>. In his work, he defined his infamous Basic Law V as:<math display="block">\varepsilon F = \varepsilon G \equiv \forall x (F(x) \equiv G(x) ) </math>Stating that if two predicates have the same extensions (they are satisfied by the same set of objects) then they are logically equivalent, however, it was determined later that this axiom led to Russell's paradox. The first explicit statement of the modern axiom of extensionality was in 1908 by Ernst Zermelo in a paper on the well-ordering theorem, where he presented the first axiomatic set theory, now called Zermelo set theory, which became the basis of modern set theories. The specific term for "Extensionality" used by Zermelo was "Bestimmtheit". The specific English term "extensionality" only became common in mathematical and logical texts in the 1920s and 1930s, particularly with the formalization of logic and set theory by figures like Alfred Tarski and John von Neumann.

In ZF set theory

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

:<math>\forall x\forall y \, [\forall z \, (\left.z \in x\right. \leftrightarrow \left. z \in y\right.) \rightarrow x=y]</math>

or in words:

:If the sets <math>x</math> and <math>y</math> have the same members, then they are the same set. the original paper of NF, the definition D8 defines as shorthand for . This definition is based more on intension rather than extension, as it can be read as "two objects are equal if one belongs to all sets that the other belongs to (i.e., has all the properties that the other has)". This definition, as well as a variant that replaces the conditional by the biconditional, was common in Quine's time. which is logically equivalent to the ZF axiom of extensionality.

In his Mathematical Logic (1951), Quine defines as (definition D10)., exactly equivalent to the antecedent of the ZF axiom of extensionality. This is based on the principle that "classes are the same when their members are the same", although Quine seems to have taken this principle for granted at this point and does not explicitly discuss "extensionality". This definition change was motivated by a desire to be compatible with proper classes.

In ZU set theory

In the Scott–Potter (ZU) set theory, the "extensionality principle" <math>( \forall x ) (\left.x \in a\right. \Leftrightarrow \left. x \in b\right.) \Rightarrow a=b</math> is given as a theorem rather than an axiom, which is proved from the definition of a "collection".

In set theory with ur-elements

An ur-element is a member of a set that is not itself a set.

In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory.

Ur-elements can be treated as a different logical type from sets; in this case, <math>B \in A</math> makes no sense if <math>A</math> is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require <math>B \in A</math> to be false whenever <math>A</math> is an ur-element.

In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set.

To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

:<math>\forall A \, \forall B \, ( \exists X \, (X \in A) \implies [ \forall Y \, (Y \in A \iff Y \in B) \implies A = B ] \, ).</math>

That is:

:Given any set A and any set B, if A is a nonempty set (that is, if there exists a member X of A), then if A and B have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to define <math>A</math> itself to be the only element of <math>A</math>

whenever <math>A</math> is an ur-element. Such a set <math>A</math> is known as a Quine atom. While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

Notes

References

Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. .